Three numbers $a , 0 , b$ form an arithmetic sequence in this order, and three numbers $2 b , a , - 7$ form a geometric sequence in this order. What is the value of $a$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with nonzero common difference, the three terms $a _ { 2 } , a _ { 4 } , a _ { 9 }$ form a geometric sequence with common ratio $r$ in this order. Find the value of $6r$. [4 points]

Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

An arithmetic sequence $\left\{ a _ { n } \right\}$ with first term a natural number and common difference a negative integer, and a geometric sequence $\left\{ b _ { n } \right\}$ with first term a natural number and common ratio a negative integer, satisfy the following conditions. Find the value of $a _ { 7 } + b _ { 7 }$. [4 points] (가) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + b _ { n } \right) = 27$ (나) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + \left| b _ { n } \right| \right) = 67$ (다) $\sum _ { n = 1 } ^ { 5 } \left( \left| a _ { n } \right| + \left| b _ { n } \right| \right) = 81$

19. Let the common difference of an arithmetic sequence be d, the sum of the first n terms be, the common ratio of a geometric sequence be q. Given $= - = 2 , \mathrm { q } = \mathrm { d } , $ $= 100$. (I) Find the general term formulas of the sequences and. (II) When $\mathrm { d } > 1$, let $= c _ { n } = \frac { a _ { n } } { b _ { n } }$. Find the sum of the first n terms of the sequence.

18. Given that $\{ a _ { n } \}$ is a geometric sequence with all positive terms, $\{ b _ { n } \}$ is an arithmetic sequence, and $a _ { 1 } = b _ { 1 } = 1$, $b _ { 2 } + b _ { 3 } = 2 a _ { 3 }$, $a _ { 5 } - 3 b _ { 2 } = 7$.
(1) Find the general term formulas for $\{ a _ { n } \}$ and $\{ b _ { n } \}$;
(2) Let $c _ { n } = a _ { n } b _ { n } , n \in \mathbb{N} ^ { * }$. Find the sum of the first $n$ terms of the sequence $\{ c _ { n } \}$.

10. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$. If $a _ { 2 } , a _ { 3 } , a _ { 7 }$ form a geometric sequence, and $2 a _ { 1 } + a _ { 2 } = 1$ , then $a _ { 1 } =$ $\_\_\_\_$ , $d =$ $\_\_\_\_$.

3. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$, and the sum of the first $n$ terms is $S _ { n }$. If $a _ { 3 } , a _ { 4 } , a _ { 8 }$ form a geometric sequence, then
A. $a _ { 1 } d > 0 , d S _ { n } > 0$
B. $a _ { 1 } d < 0 , d S _ { n } < 0$
C. $a _ { 1 } d > 0 , d S _ { n } < 0$
D. $a _ { 1 } d < 0 , d S _ { n } > 0$ [Figure]

18. (12 points)
Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with all positive terms, $a _ { 1 } = 2 , a _ { 3 } = 2 a _ { 2 } + 16$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $b _ { n } = \log _ { 2 } a _ { n }$, find the sum of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$.

Let $b _ { i } > 1$ for $i = 1,2 , \ldots , 101$. Suppose $\log _ { e } b _ { 1 } , \log _ { e } b _ { 2 } , \ldots , \log _ { e } b _ { 101 }$ are in Arithmetic Progression (A.P.) with the common difference $\log _ { e } 2$. Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { 101 }$ are in A.P. such that $a _ { 1 } = b _ { 1 }$ and $a _ { 51 } = b _ { 51 }$. If $t = b _ { 1 } + b _ { 2 } + \cdots + b _ { 51 }$ and $s = a _ { 1 } + a _ { 2 } + \cdots + a _ { 51 }$, then
(A) $s > t$ and $a _ { 101 } > b _ { 101 }$
(B) $s > t$ and $a _ { 101 } < b _ { 101 }$
(C) $s < t$ and $a _ { 101 } > b _ { 101 }$
(D) $s < t$ and $a _ { 101 } < b _ { 101 }$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a _ { 1 } = b _ { 1 } = c$, then the number of all possible values of $c$, for which the equality
$$2 \left( a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \right) = b _ { 1 } + b _ { 2 } + \cdots + b _ { n }$$
holds for some positive integer $n$, is $\_\_\_\_$

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:
(1) $\frac{8}{5}$
(2) $\frac{4}{3}$
(3) $1$
(4) $\frac{7}{4}$

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is: (1) $\frac{8}{5}$ (2) $\frac{4}{3}$ (3) $1$ (4) $\frac{7}{4}$

If the arithmetic mean of two numbers $a$ and $b , a > b > 0$, is five times their geometric mean, then $\frac { a + b } { a - b }$ is equal to:
(1) $\frac { 7 \sqrt { 3 } } { 12 }$
(2) $\frac { 3 \sqrt { 2 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 2 }$
(4) $\frac { 5 \sqrt { 6 } } { 12 }$

Let $a, b$ and $c$ be the $7^{\text{th}}, 11^{\text{th}}$ and $13^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to:
(1) 2
(2) $\frac{7}{13}$
(3) $\frac{1}{2}$
(4) 4

If three distinct numbers $a , b , c$ are in G.P. and the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?
(1) $\frac { d } { a } , \frac { e } { b } , \frac { f } { c }$ are in A.P.
(2) $d , e , f$ are in A.P.
(3) $d , e , f$ are in G.P.
(4) $\frac { d } { a } , \frac { e } { b } , \frac { f } { c }$ are in G.P.

If $p$ and $q$ are real numbers such that $p + q = 3 , p ^ { 4 } + q ^ { 4 } = 369$, then the value of $\left( \frac { 1 } { p } + \frac { 1 } { q } \right) ^ { - 2 }$ is equal to (if the full expression were available).

Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^ { 2 } - 8ax + 2a = 0$ and $q$ and $s$ are the roots of the equation $x ^ { 2 } + 12bx + 6b = 0$, such that $\frac { 1 } { p }, \frac { 1 } { q }, \frac { 1 } { r }, \frac { 1 } { s }$ are in A.P., then $a ^ { - 1 } - b ^ { - 1 }$ is equal to $\_\_\_\_$.

If $8 = 3 + \frac { 1 } { 4 } ( 3 + p ) + \frac { 1 } { 4 ^ { 2 } } ( 3 + 2 p ) + \frac { 1 } { 4 ^ { 3 } } ( 3 + 3 p ) + \ldots \infty$, then the value of $p$ is

Let three real numbers $a , b , c$ be in arithmetic progression and $a + 1 , b , c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a , b$ and $c$ is 8 , then the cube of the geometric mean of $a , b$ and $c$ is
(1) 128
(2) 316
(3) 120
(4) 312

A geometric sequence $(b_n)$ with first two terms $b_{1} = \frac{4}{3}$ and $b_{2} = 2$ and an arithmetic sequence $(a_n)$ whose common difference equals the common ratio of this geometric sequence are given.
If $b_{7} = a_{11}$, what is $a_{1}$?
A) $\frac{1}{4}$ B) $\frac{1}{8}$ C) $\frac{3}{8}$ D) $\frac{3}{16}$ E) $\frac{5}{16}$